Hexagon functions and the three-loop remainder function

By L. Dixon, J. Drummond, M. Hippel, Jeffrey Pennington. 2013

187 citations recorded by the source corpus.

A bstractWe present the three-loop remainder function, which describes the scattering of six gluons in the maximally-helicity-violating configuration in planar $ \mathcal{N} $ = 4 super-Yang-Mills theory, as a function of the three dual conformal cross ratios. The result can be expressed in terms of multiple Goncharov polylogarithms. We also employ a more restricted class of hexagon functions which have the correct branch cuts and certain other restrictions on their symbols. We classify all the hexagon functions through transcendental weight five, using the coproduct for their Hopf algebra iteratively, which amounts to a set of first-order differential equations. The three-loop remainder function is a particular weight-six hexagon function, whose symbol was determined previously. The differential equations can be integrated numerically for generic values of the cross ratios, or analytically in certain kinematic limits, including the near-collinear and multi-Regge limits. These limits allow us to impose constraints from the operator product expansion and multi-Regge factorization directly at the function level, and thereby to fix uniquely a set of Riemann ζ valued constants that could not be fixed at the level of the symbol. The near-collinear limits agree precisely with recent predictions by Basso, Sever and Vieira based on integrability. The multi-Regge limits agree with the factorization formula of Fadin and Lipatov, and determine three constants entering the impact factor at this order. We plot the three-loop remainder function for various slices of the Euclidean region of positive cross ratios, and compare it to the two-loop one. For large ranges of the cross ratios, the ratio of the three-loop to the two-loop remainder function is relatively constant, and close to −7.

Subfields: Nuclear and High Energy Physics

View Paper: https://www.semanticscholar.org/paper/c9658e51bdc43a525d3fc2e3c01083fd2f357c17

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In Nuclear and High Energy Physics

1. L. Dixon
2. Claude Duhr
3. P. Vieira
4. M. Spradlin
5. A. Sever
6. B. Basso
7. J. Drummond
8. C. Cheung

Citations in the corpus

• A symbol of uniqueness: the cluster bootstrap for the 3-loop MHV heptagon (2014, methodology): “[60, 61], leading to further recursive equations, relevant for determining non-MHV amplitudes. The hexagon bootstrap programme has yielded explicit expressions up to four loops for the MHV amplitudes [16, 18, 19] and three loops in the NMHV case [17, 21]. For higher multiplicities, explicit results ”
• Heptagons from the Steinmann cluster bootstrap (2016, methodology): “will also have to be determined. On the other hand, sometimes the function-level constraints are more powerful than the symbol-level constraints. As rst observed in the case of the 3-loop MHV hexagon [10,30], the number of n-gon functions obeying additional constraints, such as well-dened collinear ”
• PolyLogTools — polylogs for the masses (2019, background): “0] for a pedagogical introduction of how to derive identities among MPLs using the coproduct. Another important application of the coproduct is the amplitude bootstrap in planar N= 4 Super Yang-Mills [21{24,26,51]. Here we will not discuss these applications in detail, but we focus on the implementa”
• A periodic table of effective field theories (2016, methodology): “to alternative formulations of the S-matrix, for instance using recursion relations [1{3], unitarity methods [4,5], and more recently scattering equations [6{8], BCJ duality [9{11], hexagon bootstrap [12,13], ux tube S-matrix [14,15], twistor methods [16{24], Grassmannians [25], on-shell diagrams an”
• Space-time S-matrix and flux-tube S-matrix III. The two-particle contributions (2014, methodology): “Instead, we can rely on the alternative approach proposed by Dixon and collaborators for bootstrapping these objects at remarkably high number of loops [7,8].”